Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{10p^{-2}y^{5}}{-2(p^{4}y^{2})^{4}}\\)

Explanation:

Step1: Simplify the coefficient and apply power of a product rule

First, simplify the coefficient: $\frac{10}{-2} = -5$. Then, apply the power of a product rule $(ab)^n = a^n b^n$ to $(p^4 y^2)^4$: $(p^4)^4 (y^2)^4 = p^{16} y^8$. So the expression becomes $\frac{-5 p^{-2} y^5}{p^{16} y^8}$.

Step2: Apply the quotient rule for exponents

The quotient rule for exponents is $\frac{a^m}{a^n} = a^{m - n}$. For the $p$ terms: $\frac{p^{-2}}{p^{16}} = p^{-2 - 16} = p^{-18}$. For the $y$ terms: $\frac{y^5}{y^8} = y^{5 - 8} = y^{-3}$. Now the expression is $-5 p^{-18} y^{-3}$.

Step3: Convert negative exponents to positive exponents

Recall that $a^{-n} = \frac{1}{a^n}$. So $p^{-18} = \frac{1}{p^{18}}$ and $y^{-3} = \frac{1}{y^3}$. Substituting these in, we get $-5 \cdot \frac{1}{p^{18}} \cdot \frac{1}{y^3} = -\frac{5}{p^{18} y^3}$.

Answer:

$-\dfrac{5}{p^{18} y^3}$